Fixed Point Theory, Volume 5, No. 2, 24, 181-195 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.htm FIXED POINT THEOREMS OF KRASNOSELSKII TYPE IN A SPACE OF CONTINUOUS FUNCTIONS CEZAR AVRAMESCU 1 AND CRISTIAN VLADIMIRESCU 2 1 Centre for Nonlinear Analysis and Its Applications University of Craiova A. I. Cuza Street, No. 13, Craiova RO-2585, Romania E-mail: zarce@central.ucv.ro 2 Department of Mathematics University of Craiova A. I. Cuza Street, No. 13, Craiova RO-2585, Romania E-mail: vladimirescucris@hotmail.com Abstract. A well known result due to Krasnoselskii ensures the existence of a fixed point to an operator K = A + B which is defined on a closed convex and bounded subset of a Banach space X, where A is a contraction operator and B is a compact operator. In the particular case when X = C J, IR N, the condition B is compact can be replaced by a weaker one, of Lipschitz type in an integral form. In the present Note, on a closed convex and bounded subset of the space X = C J, IR N J IR being a compact or not compact interval) one considers an operator K = A + B + C, where A and B fulfill the conditions mentioned above and C is a compact operator. To this operator K, certain theorems of existence of at least one fixed point are presented and some particular cases are distinguished. Key Words and Phrases: Fixed points, spaces of continuous functions. 2 Mathematics Subject Classification: 47H1, 45G1. 1. Introduction Two main results of the fixed point theory are Schauder s theorem and the Banach Contraction Principle. Krasnoselskii combined them into the following result see [9]). This paper was presented at International Conference on Nonlinear Operators, Differential Equations and Applications held in Cluj-Napoca Romania) from August 24 to August 27, 24. 181
182 CEZAR AVRAMESCU AND CRISTIAN VLADIMIRESCU Theorem 1.1. Let M be a closed convex and bounded subset of a Banach space X. Suppose that A and C map M into X such that: 1) A is α contraction with α < 1; 2) C is compact operator; 3) Ax + Cy M, x, y M. Then there exists y M such that y = Ay + Cy. 1.1) Remark that C is a compact operator if it is continuous and CM is a compact subset of X, for every bounded set M.) This result is captivating and it has many interesting applications. The proof idea consists in the fact that hypothesis 1) ensures the existence and the continuity to operator I A) 1 where I denotes the identity operator). Then the solutions to 1.1) coincide with the fixed points to operator I A) 1 C; but the operator I A) 1 C : M M is a compact one and so, due to Schauder s theorem, the existence of a fixed point to this operator is assured. Krasnoselskii s theorem has known different generalizations and improvements see, e.g., [5], [6], [15]). In [5], Burton proved the following improvement of Theorem 1.1. Theorem 1.2. Let M be a closed convex and bounded subset of a Banach space X. Suppose that the following conditions are fulfilled: i) A : X X is contraction with constant α < 1; ii) C : M X is compact; iii) x = Ax + Cy, y M) = x M). Then 1.1) has solutions. In [6] another result of Krasnoselskii type is obtained through the following theorem of Schaefer see, e.g., [14]): Theorem 1.3. Let E be a locally convex space and H : E E compact operator. Then either S 1 ) the equation x = λhx has a solution for λ = 1, or S 2 ) the set {x E, x = λhx, λ, 1)} is unbounded.
FIXED POINT THEOREMS OF KRASNOSELSKII TYPE 183 Using this result, Burton & Kirk see [6]) proved the following theorem. Theorem 1.4. Let X be a Banach space, A : X X contraction with constant α < 1, and C : X X compact operator. Then either the equation x = λa x/λ) + λcx has a solution for λ = 1 or the set {x X, x = λa x/λ) + λcx, λ, 1)} is unbounded. The proof is based on the fact that the operator x λa x/λ) is a contraction with constant α and therefore ) x = λa x/λ) + λcx) x = λ I A) 1 Cx. The method used by Burton & Kirk is used by Dhage [7], [8]) in the case when A is a nonlinear contraction or A is a compact operator having the property that there exists a p 1 such that A p is a contraction with constant α < 1. In our Note [1], we obtained results of Krasnoselskii type in the case when X is a Fréchet space and in [2] we deduced results of Krasnoselskii type to an operator of the form x, y) A x, y), C x, y)), where A, y) is a contraction with α < 1 for every y and C x, ) is a compact operator for every x. Although the fixed point theorems are stated in Banach spaces or Fréchet spaces) X, the more frequent cases we can meet in applications are those when X is a space of continuous functions or integrable functions respectively locally integrable in the case of the Fréchet spaces). In this particular case, the hypotheses become often more general. Recently, other results of Krasnoselskii type have been obtained in the particular case X = C J, IR N), where J is a compact or not compact interval. In [15] one proves the existence of a unique solution to equation x = Ax + Bx 1.2) in the case when X = C [, T ], IR N), A, B : X X satisfy the conditions Ax) t) Ay) t) α x t) y t), t [, T ], x, y X, 1.3) where α [, 1), Bx) t) By) t) b x s) y s) ds, t [, T ], x, y X, 1.4)
184 CEZAR AVRAMESCU AND CRISTIAN VLADIMIRESCU where b >, and denotes a norm in IR N. Remark that in [15] the more restrictive condition α [, 1/e) is used, but it still holds for α < 1. The operators A, B, of this type, arise in the study of mathematical models which describe the contact between a deformable body and a foundation. The same problem is independently considered in [1], [11], wherein condition 1.4) is replaced by Bx) t) By) t) b t β x s) y s) ds, b >, t [, T ], 1.5) where β [, 1). In [3] this result is extended to the case X = C IR +, IR N), where IR + := [, ); in the same manner the result can be obtained in the case when X = C IR, IR N). The proof of the results mentioned is based on the fact that, in the conditions stated above, the operator D = A + B is contraction in X, eventually by using a norm convenient and equivalent to the classical norm. In the present Note we shall research the existence of the fixed points in X = C J, IR N), where J is a compact or not compact interval, to operators of type K = A + B + C, 1.6) where A and B satisfy conditions of type 1.3), 1.5), and C is a compact operator. The first result is proved by using Schauder-Tychonoff theorem and the second by using Burton-Kirk theorem cited. Next we prove a theorem of existence of the fixed points to a product mapping of type x, y) U x, y), V x, y)). The last existence result is proved in the case when K is a multi-valued operator. Finally, in the last section, we present applications concerning the proof of the existence of the solutions to integral equations with modified argument. 2. Main results 2.1. General hypotheses and notations. In the sequel, denotes a norm in IR N. Let J IR be an interval, where J = [, T ] or J = IR + the case J = IR is treated similarly). Set X := C J, IR N) = { x : J IR N, x continuous }.
FIXED POINT THEOREMS OF KRASNOSELSKII TYPE 185 If J = [, T ], then X is a Banach space equipped with the norm of the uniform convergence on J, A norm equivalent to which will be used is where x := sup { x t) }. 2.1) t J x = x γ + x h, 2.2) { } x γ = sup t γ { x t) }, x h = sup γ t T e ht γ) x t), γ, T ) and h > is an arbitrary number. If J = IR +, then X can be organized as a Fréchet space i.e., a linear topological metrisable and complete space) endowed with the numerable family of seminorms x n := sup { x t) }, n IN\{}. 2.3) t [,n] The Banach Contraction Principle can be easily extended to an arbitrary Fréchet space. Definition 2.1. Let E be a Fréchet space and M E an arbitrary subset. The function D : M E is said to be a contraction if there exists a family of seminorms { x n } n IN\{}, equivalent to the initial family of E) and for every n IN\{} there exists α n < 1 such that Dx Dy n α n x y n, x, y M, n IN\{}. 2.4) Proposition 2.1. Banach) Let E be a Fréchet space, M E a closed subset, and D : M M a contraction. Then D admits a unique fixed point and the operator I D admits a continuous inverse. The proof of Proposition 2.1 is classical and it uses the successive approximations scheme. In addition recall that Schauder s fixed point theorem works as well as in Fréchet spaces; in this case it is known as the Schauder-Tychonoff theorem. In what follows we consider the following functions with the specified properties:
186 CEZAR AVRAMESCU AND CRISTIAN VLADIMIRESCU a : J [, c), c < 1, b : J IR + bounded function, ν, σ : J J continuous functions with ν t) t, σ t) t, t J. Let M X be an arbitrary subset. Consider that the operators A, B, C : M X fulfil the following hypotheses: A) Ax) t) Ay) t) a t) x ν t)) y ν t)), t J, x, y M; B) Bx) t) By) t) bt) t β x σ s)) y σ s)) ds, t J, x, y M, where β [, 1); C) C : M X is compact operator. In the case a t) α, b t) b, ν t) = σ t) = t, J = [, T ] hypotheses A), B) are identical to 1.3), 1.5). Set K := A + B + C, D := A + B. 2.2. The first existence result. The first existence result is contained in the following theorem. Theorem 2.1. Suppose that: i) M is a closed convex and bounded set; ii) A, B, C fulfil hypotheses A), B), C); iii) for every x, y M one has Dx + Cy M. Then K admits at least one fixed point. Proof. It is enough to prove Theorem 2.1 in the case J = [, T ], since in the case J = IR + the reason repeats on each compact [, n], n IN\{}. The key of the proof consists in showing that D is contraction; then, ) x = Kx) x = I D) 1 Cx and so the conclusion follows from an easy application of Schauder-Tychonoff theorem to I D) 1 C on M. We shall apply the Banach Contraction Principle. To this aim, we show that D is contraction, i.e. there exists δ [, 1) such that for any x, y M, Dx Dy δ x y.
FIXED POINT THEOREMS OF KRASNOSELSKII TYPE 187 Let t [, γ] be arbitrary. Then we have and hence Dx) t) Dy) t) a t) x ν t)) y ν t)) Dx Dy γ where c 1 := sup t J {b t)}. Let t [γ, T ] be arbitrary. Then we get + b t) t β x σ s)) y σ s)) ds a t) + b t) γ 1 β) x y γ c + c 1 γ 1 β) x y γ, 2.5) Dx) t) Dy) t) a t) x ν t)) y ν t)) + b t) γ x σ s)) y σ s)) ds + t β After easy estimates, it follows that γ ) x σ s)) y σ s)) e hσs) γ) e hσs) γ) ds. Dx) t) Dy) t) e ht γ) < a t) x ν t)) y ν t)) e ht γ) +b t) γ 1 β x y γ + b t) h γ β x y h and therefore Dx Dy h c sup { x ν t)) y ν t)) e ht γ)} t [γ,t ] +c 1 γ 1 β x y γ + c 1 h γ β x y h c + c 1 h γ β) x y h + c 1 γ 1 β x y γ. 2.6) By 2.5) and 2.6) we obtain Dx Dy c + 2c 1 γ 1 β) x y γ + Since c [, 1), for γ c + c 1 h γ β) x y h. 2.7) ) 1 ), 1 c 1 β 2c 1 we deduce c + 2c 1 γ 1 β < 1 and for h > c 1 1 c γ β we deduce c+ c 1 h γ β < 1. Let δ := max { c + 2c 1 γ 1 β, c + c 1 h γ β}. It follows that δ < 1 and, since 2.7), ) Dx Dy δ x y γ + x y h = δ x y.
188 CEZAR AVRAMESCU AND CRISTIAN VLADIMIRESCU Hence, D is contraction. Remark 2.1. If C we get the results from [1], [11]. If B the Krasnoselskii s theorem is deduced in this particular case C J, IR N). 2.3. The second existence result. In this subsection we consider the case M = X. Theorem 2.2. Admit that hypotheses A), B), C) are fulfilled and that D) the set {x X, x = λd x/λ) + λcx, λ, 1)} is bounded. Then K admits at least one fixed point. Proof. The proof is reduced to the remark that ) x = λd x/λ) + λcx) x = λ I D) 1 Cx and to an application of Burton & Kirk Theorem 1.4. 2.4. The third existence result. In this subsection we consider the problem of the existence of the solutions to a system of type { x = A 1 x, y) + B 1 x, y) 2.8) y = A 2 y) + B 2 x, y), where A 1, B 1, B 2 : M X, A 2 : M 2 X, X = C J, IR N), J = [, T ], M = M 1 M 2, M 1 X is a closed and bounded set, M 2 X is a closed convex and bounded set. The case X = C IR +, IR N) is treated similarly.) Consider the following hypotheses: 1) A 1 x 1, y) t) A 1 x 2, y) t) α y) x 1 ν t)) x 2 ν t)), x 1, y), x 2, y) M, t J, where α : M 2 [, 1), and A 1 x, y) is continuous with respect to y, for every x M 1 ; 2) B 1 x 1, y) t) B 1 x 2, y) t) by) t βy) x 1 σ s)) x 2 σ s)) ds, t J, x 1, y), x 2, y) M, where b : M 2, ), β : M 2 [, 1), and B 1 x, y) is continuous with respect to y, for every x M 1 ; 3) A 2 y 1 ) A 2 y 2 ) L y 1 y 2, where L [, 1) is constant; 4) B 2 : M X is a compact operator; 5) x, y) M, A 1 x, y) + B 1 x, y) M 1 ; 6) y M 2, x, v) M, A 2 y) + B 2 x, v) M 2. We can state and prove the following result.
FIXED POINT THEOREMS OF KRASNOSELSKII TYPE 189 Theorem 2.3. Suppose that hypotheses 1) 6) are fulfilled. Then system 2.8) admits solutions. Proof. Let y M 2 be arbitrary fixed. By hypotheses 1), 2), an argument similar to the one from the proof of Theorem 2.1 leads us to conclude that the operator x A 1 x, y) + B 1 x, y) is a contraction. From hypothesis 5) we conclude that this operator maps M 1 into M 1. Therefore, by applying the Banach Contraction Principle, we deduce that there exists g y) M 1 such that g y) = A 1 g y), y) + B 1 g y), y). First one shows that the mapping g : M 2 M 1 is continuous. operator So, the y B 2 g y), y) is continuous. By hypothesis 4) we get that this operator is compact from M 2 to X. Therefore, using hypotheses 3), 6), and applying Krasnoselskii s Theorem 1.1, we conclude that the operator y A 2 y) + B 2 g y), y) admits fixed points. The proof of Theorem 2.3 is now complete. 2.5. Multivalued operators. In the excellent book of A. Petruşel [12] the author presents certain theorems of Krasnoselskii type in the case of the multivalued operators see Theorems 2.8.4, 2.8.5, 2.8.6). Using these results in the particular case of the space C [, T ], IR N), we can obtain a fixed point theorem for the sum of three operators. It is necessarily first to make some notations. So, let X = C [, T ], IR N). Set P b,cl X) : = {Y X, Y, Y bounded and closed}, P cp,cv X) : = {Y X, Y, Y compact and convex}, P b,cl,cv X) : = P b,cl X) P cp,cv X).
19 CEZAR AVRAMESCU AND CRISTIAN VLADIMIRESCU Let M P cp,cv X). Consider the operators A, B : M P b,cl,cv X), C : M P cp,cv X). Theorem 2.4. Suppose that the following hypotheses are fulfilled: i) x i M, y i A x i ), i = 1, 2, the following inequality holds: y 1 t) y 2 t) a x 1 t) x 2 t), t [, T ], a [, 1); ii) x i M, y i B x i ), i = 1, 2, the following inequality holds: y 1 t) y 2 t) b t β x 1 s) x 2 s) ds, b >, β [, 1); iii) C is l.s.c. and compact operator; iv) A x) + B x) + C y) M, x M, y M. Then, F ix A + B + C). Proof. Consider on M the metric d x, y) := x y γ + x y h. As in the proof of Theorem 2.1 one can show that the operator D := A + B is contraction with respect to the Hausdorff-Pompeiu metric H, generated by d. By using Theorem 2.8.4 from [12], the conclusion follows. Remark 2.2. Hypothesis iv) can be replaced by the condition of Burton type: if y D y) + C x), x M, then y M. Remark 2.3. If M P b,cl,cv X) and A, B, C : M P cp,cv X), then C can be u.s.c. and compact. 3. Applications 3.1. The first application. In this subsection we shall present an application of Theorem 2.2 to establish the existence of solutions to an integral equation with modified argument of type x t) = f t) + F t, x ν t))) + + εt) ρt) G t, s, x σ s))) ds Φ t, s) Ψ s, x µ s))) ds, 3.1) admitting that the following hypotheses are fulfilled: f C J, IR N) ; 3.2)
FIXED POINT THEOREMS OF KRASNOSELSKII TYPE 191 F : J IR N IR N is a continuous and bounded function; 3.3) where a : J [, 1) ; F t, x) F t, y) a t) x y, t J, x, y IR N, 3.4) G : IR N IR N is a continuous function, 3.5) where := {s, t) J J, s t} ; G t, s, x) G t, s, y) g t, s) x y, x, y IR N, s, t) ; 3.6) G t, s, ) ; 3.7) Φ : M N IR) is a continuous quadratic matrix; 3.8) Ψ : J IR N IR N is a continuous function; 3.9) Ψ t, x) ϕ t) ψ x ), 3.1) where ϕ : J IR + is continuous, ψ : IR + IR + is continuous and increasing, and ν, σ, µ : J J are continuous functions such that ν t) t, σ t) t, µ t) t, t J. Denote Ax) t) := f t) + F t, x ν t))), Bx) t) := ρt) G t, s, x σ s))) ds, Cx) t) := εt) Φ t, s) Ψ s, x µ s))) ds. It is easily seen that conditions A), B), C) from subsection 2.1 are fulfilled with β =. In order to apply Theorem 2.2 it will be sufficient to check condition D) from subsection 2.3. To this purpose, it is enough to check it in the case J = [, T ]; for the case J = IR + the reason will repeat on each interval [, n], n IN\{}. So, let λ, 1) and x C J, IR N), such that x t) = λf t) + λf +λ εt) t, 1λ x ν t)) ) + λ ρt) G t, s, 1λ ) x σ s)) ds + Φ t, s) Ψ s, x µ s))) ds. 3.11) First, from 3.2) and 3.3) there exists a C 1 >, such that t, λf t) + λf 1 ) λ x ν t)) C 1, t [, T ].
192 CEZAR AVRAMESCU AND CRISTIAN VLADIMIRESCU Next, since we get λ ρt) G t, s, x) = G t, s, x) G t, s, ) g t, s) x, G t, s, 1λ ) x σ s)) ds where C 2 := sup s t T {g t, s)}; εt) λ Φ t, s) Ψ s, x µ s))) ds where C 3 := sup s t T { Φ t, s) ϕ t)}. In conclusion, we obtain from 3.11) ρt) g t, s) x σ s)) ds C 2 x σ s)) ds, Φ t, s) ϕ t) ψ x µ s)) ) ds C 3 ψ x µ s)) ) ds, x t) C 1 + C x σ s)) ds + C 3 ψ x µ s)) ) ds, 3.12) where C := max {C 2, C 3 }. We set w t) := sup s t T { x s) }. Obviously, w is an increasing function and x t) w t), t [, T ], and therefore x σ s)) w s), x µ s)) w s), Definitively we deduce by 3.12) and 3.13) ψ x µ s)) ) ψ w s)). 3.13) w t) C 1 + C [w s) + ψ w s))] ds. 3.14) We denote u t) := C 1 + C [w s) + ψ w s))] ds. By relation 3.14) it follows that w t) u t), hence x t) u t). But u t) = C [w t) + ψ w t))] C [u t) + ψ u t))],
FIXED POINT THEOREMS OF KRASNOSELSKII TYPE 193 and, since u ) = C 1, we obtain ut) C 1 ds s + ψ s) C, t [, T ]. 3.15) Taking into account relation 3.15) we easily deduce the following corollary. Corollary 3.1. Suppose that hypotheses 3.2) 3.1) are fulfilled. Then, if ds ) s+ψs) =, equation 3.1) has at least one solution. Remark 3.1. It is clear that the condition C 1 ds s+ψs) > C. ) ds s+ψs) = can be replaced by 3.2. The second application. In this subsection we shall present an application of Theorem 2.3 in order to establish the existence of solutions to an integral system with modified argument of type { x t) = K t, s, y µ 1 s))) x ν 1 t)) ds + F t, s, x σ 1 s)), y θ 1 s))) ds y t) = T G t, s, y µ 2 s))) ds + T H t, s, x σ 2 s)), y θ 2 s))) ds, 3.16) where K : IR N M N IR) is continuous and bounded quadratic matrix, F : IR N IR N IR N, G, H : J J IR N IR N are continuous and bounded functions, such that: F t, s, u 1, v) F t, s, u 2, v) L 1 u 1 u 2, 3.17) t, s, u 1, v), t, s, u 2, v) IR N IR N, t, s, v 1 ), t, s, v 2 ) J J IR N, G t, s, v 1 ) G t, s, v 2 ) L 2 v 1 v 2, 3.18) H t, s, u, v 1 ) H t, s, u, v 2 ) L 3 v 1 v 2, 3.19) t, s, u, v 1 ), t, s, u, v 2 ) J J IR N IR N, ν 1, σ 1, µ 1, θ 1, σ 2, µ 2, θ 2 : J J are continuous functions such that ν 1 t) t, σ 1 t) t, µ 1 t) t, θ 1 t) t, σ 2 t) t, µ 2 t) t, θ 2 t) t, t J, and L 1, L 2, L 3 [, 1) are constant. Consider X = C [, T ], IR N), J = [, T ], M = B ρ B ρ, M 1 = M 2 = B ρ, where B ρ := {z X, z ρ}, ρ >,
194 CEZAR AVRAMESCU AND CRISTIAN VLADIMIRESCU and denote a : = sup t,s,v) IR N { K t, s, v) }, b : = sup t,s,u,v) IR N IR N { F t, s, u, v) }, c : = sup t,s,v) J J IR N { G t, s, v) }, d : = sup t,s,u,v) J J IR N IR N { H t, s, u, v) }. Let us define the operators A 1, B 1, B 2 : M X, A 2 : M 2 X, through A 1 x, y) t) : = B 1 x, y) t) : = A 2 y) t) : = B 2 x, y) t) : = T T K t, s, y µ 1 s))) x ν 1 t)) ds, F t, s, x σ 1 s)), y θ 1 s))) ds, G t, s, y µ 2 s))) ds, H t, s, x σ 2 s)), y θ 2 s))) ds, x, y) M, t J. In order to apply Theorem 2.3 it will be sufficient to check hypotheses 1)-6) from subsection 2.4. We shall check them in the case J = [, T ] ; for the case J = IR + the reason will repeat on each interval [, n], n IN\{}. Indeed, taking into account relations 3.17) 3.19), hypotheses 1)-6) are fulfilled with α y) at, b y) L 1, β y), L = cl 2 T, for a < 1/T, c < min {1/ L 2 T ), ρ/t }, b < ρ 1 at ) /T, and d < ρ/t c. Then, we deduce the following result. Corollary 3.2. Suppose that hypotheses 3.17) 3.19) are fulfilled. Then, if a < 1/T, c < min{1/ L 2 T ), ρ/t }, b < ρ 1 at ) /T, and d < ρ/t c, system 3.16) admits solutions. References [1] C. Avramescu, Some remarks on a fixed point theorem of Krasnoselskii, Electronic Journal of Qualitative Theory of Differential Equations, 523), 1-15. [2] C. Avramescu, C. Vladimirescu, Some remarks on Krasnoselskii s fixed point theorem, Fixed Point Theory, 423), 3-14.
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